Status: Expert contribution (submitted presently to AFNOR as NB-contribution: this contribution deals with the ISO project n°1.21.63; this project concerns a Standard for the Conceptual Schema Modelling Facilities or the CSMF standard). Records: ss-50915.doc: MS-DOS 6.0.; Window 3.1; Word 6.0. - *.txt: MS-DOS
1. NEEDS FOR MODELLING: Aspects of system engineering System designers and users, in any organisation, have skills normally covering fields other than information technology; these fields constitute their Universe of Discourse (UoD). Consequently, if information technology is of unquestionable importance in any enterprise, it must always be combined with other techniques implemented for the same purpose with equal importance; too radical a separation between these other techniques and information technology, more particularly with regard to software engineering, would be unrealistic. The desired organisation calls for the availability of a method and practical facilities for assistance in the use of information technology in their current activities, in particular for the specification of the functions they assign to the various devices, and specially to the computers, within the systems concerning them. The notions developed in the proposed paper, thus necessarily includes this plurality of techniques used; it is in relation with this broader acceptance of them, that the notion of open system and of integration must be considered. They also take in account the plurality of UoD such as: banking, transportation, industrial process control, research, etc. The engineer having overall responsibility for IT equipment in his enterprise can only have a general attitude with regard to the diversity of techniques and domains involved; however he must have methodical and technical aids enabling him to act as an informed interlocutor of contracting firms, which are no less diverse in their respective
System modelling for system engineering: Every engineer required to work on a system or on a component of a system, obviously needs any information existing on this item when performing his work. In addition, information concerning the rest of the system may also be useful to him. Thus, before doing any action on the system, he must consult a volume of information whose size increases with the size and with the complexity of the system. Simulation is the technique usually employed up to the present time to aid engineers in certain types of work. This technique, used for many years, long before the advent of data processing, makes use of means which are either created empirically or deduced by analogy or by the transformation of theoretical models of the items to be simulated. Data processing subsequently favoured the use of numerical simulation, since any program intended for calculation on the values of state parameters of any real item constitutes, in fact, a numerical simulation of this item. And, the purposes of a theoretical model involve aspects other than numerical simulation, even if data processing has found in this area a privileged field of application: other uses of models are possible. These uses constitute the domain of Computed Aided X where X represents different possible activities: design, maintenance, technician teaching, fault diagnostic,…
From this viewpoint, it should be noted that the notion of model has taken on a new connotation. Formerly, a model, expressed in the form of a program, was a technique intended for data processing. It has quickly become data itself, to be processed by techniques developed for aiding engineers in their activities.
Theorisation for modelling: The achievement of these objectives depends on the existence of strict modelling facilities. The proposed paper presents a theorisation underlying such facilities. The design of this theorisation involves three related formal tools: Physical concepts, Theoretical elements, and Language. The first part, Physical concepts, determines, in a natural language, an unitary system of concepts and method for producing assemblages of concepts. All the items of this approach may be modelled with theoretical elements because mathematics have the main advantage of providing strict models. The second part, Theoretical elements, proposes a structured collection of mathematical elements as a fundamental instrument for this general purpose. The third part, Language, determines a short formal etymology method for expressing the precedent concepts; for fixing their axiomatic origins or their relative positions in relation to other concepts; for allowing the comparison between them and other different existing definitions, and finally for introducing a formal definition method of description languages.
2 PHYSICAL CONCEPTS This chapter defines a consistent and reduced set of simple concepts; it must name these concepts and uses temporarily, for this design, a set of unusual words in order to elude ambiguity with existing vocabularies. Nevertheless, some other various definitions of similar concepts exist presently and imply the existence of various acceptations - roughly alike or entirely divergent - for each word. Different numerous normative documents state presently these various definitions; an annex gathers them.
Primary entities: Physical systems are made up of entities working interactively in order to achieve a common technical or scientific goal. Each of these entities is a device. Any device performs task on other entities; each of these other entities is a product. Devices and products are the components of the system. Exchanges of products between device are indispensable for the existence of interactions between device. We shall assume for the moment that a component can be seen as a device or as a product.
Static and dynamic entities: The modelling of a physical system and, step by step, of each entity composing it, generally includes two complementary parts: The time-independent inventory of the entities making up the system and their relations. This is the static description or the morphology of the system.
The inventory of rules scheduling interactions between entities: This is the dynamic description or the physiology of the system. An entity whose morphology and physiology are determined is a procedure. A procedure is a model since it is made up of two descriptions: a morphology and a physiology. It is a global model of a physical system, of a device or a product concerning all their possible occurrences. The concrete or experimental study of a physical system is a succession of observations of this system or of certain entities of which it is composed during operation. We shall call process the description of an operating case of a procedure. It is a special model of a physical system or device or product concerning only one of its possible occurrences in time.
System description: Analysis determines these top-down inventories; it is recursive: each morphology designates entities; each of these entities is also a system amenable to study and so on. The stopping of the study on an entity is imposed either by a decision on the part of the analyst or by a lack of knowledge. The reverse construction consisting in assembling entities and governing their interactions to form from bottom to up a new entity is a synthesis. Generally, the modelling of a system is a succession of analyses and syntheses.
3 THEORETICAL ELEMENTS: The theoretical elements proposed are intended to support the modelling of the preceding physical concepts.
General structure of the chapter: The first section, Primitive elements, of this part introduces the theoretical elements necessary for modelling a process and then a procedure. This approach uses the set theory and some topological structures in order to take in account the different aspects of continuity or of discontinuity of the elements. The second section, Recursive elements, constructs on the preceding elements the mathematical models of the simple components of a system. It uses for that purpose the algebra concept of category for modelling a product and the concept of functor for modelling a device. Let us note that the set theory and the category theory are both logical theories; the axioms, definitions and theorems of fundamental logic are available and therefore must be applied. The third section, Assemblies, shows how to combine categories or functors in order to represent complex entities and finally systems. These results may be used for modelling the concepts of interoperability and portability.
Overview of the first section: These elements have to do with whatever is immediately perceptible in a system: observable facts or those that intuition or accumulated experience suggest as observable. Then, a process describes a real or imagined phenomenon but only in time and space: it involves the functioning of the system or of an entity composing this system; the collection of all possible processes ruled by a procedure constitutes the fundamental space associated with this procedure. Reciprocally any process ruled by a procedure is an element of the fundamental space associated with this procedure. General laws exist for each collection of processes; translated mathematically, they allow finally the structuring of the fundamental space and its transformation into a new reduced space called the substrate space. This substrate space is also associated with the concerned procedure, the fundamental space is the scalar product of the substrate space and of the time viewed as a set.
As we are concerned with observable phenomena, intuition immediately suggests that any process should be expressed by a function allowing time and space as variables. Modelling experience however warns us about the known drawbacks of this representation method: there are techniques, and in particular signal processing, for which the well known functional representation does not readily account for certain physical properties such as, for example, discontinuities. The proposed approach thus avoids having to set a priori a precise and definitive mathematical form for the process model. We study the fundamental space without considering any particular form for this process. The properties of the fundamental space result directly or by deduction from primary hypotheses allowed for the represented entities.
Overview of the second section: Each component may be viewed as a procedure. Then, the topomogical structure of the substrate space associated with this component is used for modelling its set of states and all its changes of states. Finally we use the concept of category for modelling globally any component: the objects of the category are the states of the component and the morphisms are the changes of states. The concept of functor is then available for modelling any component performing a transformation upon any other component. As a result, category theory gives at once all the basic rules for assembling the precedent components, and for ordering them in time, and consequently for modelling systems.
Overview of the third section: This section examines more particularly the ability of the assembled components to function interactively; this problem deals with the physical consistency of the system. The mathematical model enables this property to be studied by offering the means of distinguishing its different aspects. The section examines in particular the questions of interoperability and portability, linking them with the examination of the consistency of the mathematical expressions of the entities concerned.
4 LANGUAGE: During the eighteenth century the existence of such an universal scientific language was soon supposed by Leibnitz. Unfortunately, Leibnitz papers tell us almost nothing about that. It is one of the most famous enigma of mathematics history. We can say nowadays that the binary formalism discovered or re-discovered by Leibnitz is perhaps the basis of this hypothetical language. The conclusion to which we are coming since the beginning of computers era is that in the present state of the technology, binary language is obviously at the lowest level, the only universal language with which a problem may be expressed and solved; nevertheless, in practice, it is easily readable and understandable only by computer and never by human mind. Subsequently this part reminds shortly the principles for designing any high level description language. Such a language must be readable by computer and also understandable by human mind; it must possesses without restriction the property of universality. This part shows how the theory of category may be used now for underlying the definition of this kind of languages. The main step in this approach is the definition of a primitive formalism which may be a standard. This primitive formalism may be viewed as a sound common foundation allowing the design of different final high level languages for system description.